So one of my characters adopted Runa Fair-Shield. She had found a fox some time ago and I let her keep it as a pet because foxes are awesome. However, while paying a visit to my Skyrim family recently, I noticed that when practicing with a dagger I gave her, she accidentally killed Vix the fox.

Will they eventually find a new pet they will beg me to let stay? Also, hypothetically speaking, if I were to say "no" to a mud crab or something being a pet, would they stop trying to get a pet altogether?

I think they will stop getting pets altogether if you say "Not ever" If you said just "No" then they would get another pet in the future.

On PC I actually planned the death of the pet (the skeever), by giving Blaise an enchanted weapon, and while he practices, I console moved the animal into his attack range and he killed it. After a while he got a mud-crab, which I really like much better than the skeever.

I think it depends on whether the family already had two pets, because it seems to me that is all you can have no matter what happens to them, and if there is only one child i know they can have a pet and adopt a dog as well. If you already had two pets and one died, I do not think they would ask about any more pets. But if you are still looking for a pet, saying no does not to something you do not want them to have is not going to stop them trying again.

I'm happy my kids always decided to adopt bunnies. I'd automatically kill any skeever they'd bring up home.